Finite Group Databasebeta

← #44560721

Subgroups — #44560721

Order 32128.

404 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632315C211yesmaximal, characteristic, complemented
2#30632256C211yesmaximal, characteristic
3#30632052C212yesmaximal, complemented
4#30632047C212yesmaximal, complemented
5#30632042C212yesmaximal, complemented
6#30632034C212yesmaximal, complemented
7#30631693C211yesmaximal, characteristic, complemented
8#30631689C211yesmaximal, characteristic, complemented
9#30631684C211yesmaximal, characteristic, complemented
10C2×#22548637C211yesmaximal, characteristic, complemented
11#405×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14#22548715C2212yescomplemented
15#22548714—21no—
16#22548712C2212yes—
17#22548707C2212yescomplemented
18#22548706—21no—
19#22548703C2212yes—
20#22548692—22no—
21#22548690—21no—
22#22548690—21no—
23#22548685—22no—
24#22548674C2211yescharacteristic
25C2×C4⋊Dic251C2211yescharacteristic
26#22548640C2211yescharacteristic, complemented
27#22548640C2211yescharacteristic, complemented
28#22548640C2212yescomplemented
29#22548637C2212yes—
30#22548637C2212yes—
31#22548637C2212yes—
32#22548637C2212yes—
33#22548637C2211yescharacteristic, complemented
34#22548637C2211yescharacteristic, complemented
35#22548637C2212yescomplemented
36C2×M16×C251C2211yescharacteristic
37C22×C8×C251C2211yescharacteristic
38#117×C251C2211yescharacteristic
39#97×C251C2212yescomplemented
40C4×C8×C251C2212yescomplemented
41C23×D251—21no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502—22no—
45C2×Dic502C2311yescharacteristic
46C2×D1004—21no—
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—22no—
50C2×D1004—22no—
51C2×D1004C2312yes—
52C2×D1004C2311yescharacteristic
53C2×C22⋊D251—21no—
54C22×Dic251—21no—
55C2×C4×D251—21no—
56C4⋊Dic251—21no—
57C4⋊Dic251—22no—
58C4⋊Dic251C2312yes—
59C4⋊Dic251C2311yescharacteristic
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—22no—
62Dic251⋊C4—22no—
63C22⋊Dic251—21no—
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4—22no—
66C4×Dic251—21no—
67C22×C4×C251C2311yescharacteristic
68M16×C251—22no—
69C2×C8×C251—22no—
70C2×C8×C251C2311yescharacteristic
71C2×C8×C251C2312yes—
72C2×C8×C251C2311yescharacteristic
73C4⋊C4×C251D411yescharacteristic
74C4⋊C4×C251D411yescharacteristic
75C22⋊C4×C251D412yes—
76C42×C251C2312yes—
77C22×D251—22no—
78C22×D251—21no—
79C22×D251—42no—
80C22×D251—81no—
81C22×D251—81no—
82Dic502—21no—
83Dic502—21no—
84Dic502—82no—
85D1004—22no—
86D1004—22no—
87D1004—21no—
88D1004—21no—
89D1004—42no—
90D1004—82no—
91D1004—82no—
92C22⋊D251—42no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—21no—
96C2×Dic251—22no—
97C2×Dic251—81no—
98C4×D251—42no—
99C23×C251C2×D411yescharacteristic
100C2×C4×C251C2×D412yes—
101C2×C4×C251C2×D412yes—
102C2×C4×C251C2×D411yescharacteristic
103C2×C4×C251#4112yes—
104C2×C4×C251#4112yes—
105C2×C4×C251C2411yescharacteristic
106C8×C251—21no—
107C8×C251—21no—
108C8×C251—22no—
109C22×C251—21no—
110C22×C251—21no—
111C22×C251#12412yes—
112C22×C251C22×D411yescharacteristic
113C2×D251—41no—
114C2×D251—42no—
115C2×D251—41no—
116C2×D251—42no—
117C2×D251—41no—
118C2×D251—42no—
119C2×D251—84no—
120C4×C251—22no—
121C4×C251—22no—
122C4×C251#12412yes—
123C4×C251C2×#4111yescharacteristic
124C4×C251C2×#4111yescharacteristic
125Dic251—41no—
126Dic251—42no—
127Dic251—41no—
128Dic251—41no—
129C2×C251—22no—
130C2×C251C2×#12411yescharacteristic
131C2×C251#57611yescharacteristic
132C2×C251#57711yescharacteristic
133D251—81no—
134D251—82no—
135C251#313611yescharacteristic, complemented, Sylow
136#3136—2511nomaximal, Sylow
137#544—2511no—
138#535—2511no—
139#490—2512no—
140#489—2512no—
141#488—2512no—
142#486—2512no—
143#418—2511no—
144#417—2511no—
145#416—2511no—
146C2×#102—2511no—
147#405D25111yescharacteristic, complemented
148C2×#41—2511no—
149C22×D4—2511no—
150C2×M16C2×D25111yescharacteristic
151C22×C8C2×D25111yescharacteristic
152C4⋊Q8—5021no—
153C4⋊Q8—2512no—
154#127—2512no—
155#125—5021no—
156#125—2512no—
157#124—2512no—
158C22⋊Q8—5022no—
159#121—5021no—
160#121—5021no—
161#120—5022no—
162#117—2511no—
163#117C2×D25111yescharacteristic
164C2×C4⋊C4—2511no—
165#103—2511no—
166#103—2511no—
167#103—2512no—
168#102—2512no—
169#102—2512no—
170#102—2512no—
171#102—2512no—
172#102—2511no—
173#102—2511no—
174#102—2512no—
175#97C2×D25112yescomplemented
176C4×C8C2×D25112yescomplemented
177C24—5021no—
178#41—5022no—
179#41—5022no—
180C2×Q8—5022no—
181C2×Q8—2511no—
182C2×D4—5021no—
183C2×D4—5022no—
184C2×D4—5022no—
185C2×D4—5021no—
186C2×D4—5022no—
187C2×D4—5022no—
188C2×D4—2512no—
189C2×D4—2511no—
190C22×C4—5021no—
191C22×C4—5021no—
192C22×C4C22×D25111yescharacteristic
193M16—22no—
194C2×C8—22no—
195C2×C8C22×D25111yescharacteristic
196C2×C8C22×D25112yes—
197C2×C8C22×D25111yescharacteristic
198C4⋊C4—5021no—
199C4⋊C4—5022no—
200C4⋊C4—5022no—
201C4⋊C4—5022no—
202C4⋊C4—2512no—
203C4⋊C4—2511no—
204C4⋊C4—2511no—
205C4⋊C4D100411yescharacteristic
206C4⋊C4D100411yescharacteristic
207C22⋊C4—5021no—
208C22⋊C4—5022no—
209C22⋊C4—5022no—
210C22⋊C4D100412yes—
211C42—5021no—
212C42C22×D25112yes—
213C23—20081no—
214C23—20081no—
215C23—10042no—
216C23—5022no—
217C23—5021no—
218C23C2×D100411yescharacteristic
219Q8—20082no—
220Q8—5021no—
221Q8—5021no—
222D4—20082no—
223D4—20082no—
224D4—10042no—
225D4—10042no—
226D4—5022no—
227D4—5022no—
228D4—5021no—
229D4—5021no—
230C2×C4—20081no—
231C2×C4—10042no—
232C2×C4—5021no—
233C2×C4—5021no—
234C2×C4—5021no—
235C2×C4—5022no—
236C2×C4C2×D100412yes—
237C2×C4C2×D100412yes—
238C2×C4C2×D100411yescharacteristic
239C2×C4#1686081012yes—
240C2×C4#1686081012yes—
241C2×C4C23×D25111yescharacteristic
242C8—21no—
243C8—21no—
244C8—22no—
245C22—21no—
246C22—21no—
247C22—20084no—
248C22—10041no—
249C22—10042no—
250C22—10041no—
251C22—10042no—
252C22—10041no—
253C22—10042no—
254C22#2254870312yes—
255C22C22×D100411yescharacteristic
256C4—22no—
257C4—22no—
258C4—10041no—
259C4—10042no—
260C4—10041no—
261C4—10041no—
262C4#2254870312yes—
263C4C2×#1686081011yescharacteristic
264C4C2×#1686081011yescharacteristic
265C2—22no—
266C2—20082no—
267C2—20081no—
268C2C2×#2254870311yescharacteristic
269C2#3063249211yescharacteristic
270C2#3063249911yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#405×C251C2
socleC22×C251C22×D4
prime radical#30632256C2
hypercenter#405D251
hypocenterC251#3136
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #30632315C2semidirect
3. #30632052C2semidirect
4. #30632047C2semidirect
5. #30632042C2semidirect
6. #30632034C2semidirect
7. #30631693C2semidirect
8. #30631689C2semidirect
9. #30631684C2semidirect
10. C2×#22548637C2semidirect
11. #405×C251C2semidirect
14. #22548715C22semidirect
17. #22548707C22semidirect
26. #22548640C22semidirect
27. #22548640C22semidirect
28. #22548640C22semidirect
33. #22548637C22semidirect
34. #22548637C22semidirect
35. #22548637C22semidirect
39. #97×C251C22semidirect
40. C4×C8×C251C22semidirect
135. C251#3136semidirect
147. #405D251semidirect
175. #97C2×D251semidirect
176. C4×C8C2×D251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306323152
#306322562
#306320522
#306320472
#306320422
#306320342
#306316932
#306316892
#306316842
C2×#225486372
#405×C2512
#3136251